Zorich映射逆分支上测度序列的收敛性分析

Convergence Analysis of Measure Sequences on Inverse Branches of Zorich Maps

摘要: 本文研究Zorich映射逆分支生成的紧集族上概率测度序列的弱收敛性，旨在为高维逃逸集的Hausdorff维数估计提供测度理论基础。通过构造由逆分支Λ^r生成的嵌套紧集族{N_n}，定义了支撑在迭代逆像集上的概率测度序列{μ_n}。选取合适的紧集$M\subset {ℝ}^3$ ，使得对于任意$ò$ > 0，有μ_n(M) ≥ 1 − $ò$ 对所有n成立，从而证明该测度序列的紧性。再结合Prokhorov定理，得到该概率测度的弱收敛性，并进一步以正数序列r_n(x)为桥梁建立定义在K_n(x)上的极限测度与K_n(x)直径之间的数量关系，为后续进一步估计以特定速度逃逸的点集的Hausdorff测度奠定基础。

Abstract: This paper investigates the weak convergence of sequences of probability measures supported on families of compact sets generated by inverse branches of Zorich maps, aiming to establish a measure-theoretic foundation for estimating the Hausdorff dimension of escaping sets in higher dimensions. By constructing a nested family of compact sets {N_n} induced by inverse branches Λ^r, we define a sequence of probability measures supported by iterated pre-image sets {μ_n}. Through the selection of appropriate compact sets $M\subset {ℝ}^3$ , we demonstrate that for any $ò$ > 0, there exists μ_n(M) ≥ 1 − $ò$ for all n, thereby proving the tightness of the measure sequence. Combining this with Prokhorov’s theorem, we establish the weak convergence of the probability measure sequence. Furthermore, using the positive integer sequence r_n(x) as a bridge, the quantitative relationship between the limiting measure defined on K_n(x) and the diameter of K_n(x) is established. This lays the foundation for subsequent estimation of the Hausdorff measure of sets escaping at specific rates.